Compactness and essential norms of composition operators on Orlicz–Lorentz spaces

In this work, we present necessary and sufficient conditions for compactness of the composition operator on Orlicz–Lorentz spaces and determine upper and lower estimates for the essential norm of the composition operator on these spaces.     

Sharp weighted bounds for the Hardy–Littlewood maximal operators on Musielak–Orlicz spaces

Let \({\varphi}\) be a Musielak–Orlicz function satisfying that, for any \({(x,\,t)\in{\mathbb R}^n \times [0, \infty)}\), \({\varphi(\cdot,\,t)}\) belongs to the Muckenhoupt weight class \({A_\infty({\mathbb R}^n)}\) with the critical weight exponent \({q(\varphi) \in [1,\,\infty)}\) and \({\varphi(x,\,\cdot)}\) is an Orlicz function with uniformly lower type \({p^{-}_{\varphi}}\) and uniformly upper type \({p^+_\varphi}\) satisfying \({q(\varphi) < p^{-}_{\varphi}\le p^{+}_{\varphi} < \infty}\). In this paper, the author obtains a sharp weighted bound involving \({A_\infty}\) constant for the Hardy–Littlewood maximal operator on the Musielak–Orlicz space \({L^{\varphi}}\). This result recovers the known sharp weighted estimate established by Hytönen et al. in [J. Funct. Anal. 263:3883–3899, 2012].     

On mappings in the Orlicz–Sobolev classes on Riemannian manifolds

We study the problems of the continuous and homeomorphic extension to the boundary of lower Q-homeomorphisms between domains on Riemannian manifolds and formulate the corresponding consequences for homeomorphisms with finite distortion in the Orlicz–Sobolev classes W_loc^1,j W_{loc}^{1,varphi } under a condition of the Calderon type for the function φ and, in particular, in the Sobolev classes W_loc^1,p W_{loc}^{1,p} for p > n − 1.     

Generalized fractional maximal and integral operators on Orlicz and generalized Orlicz–Morrey spaces of the third kind

Positivity - In the present paper, we will characterize the boundedness of the generalized fractional integral operators $$I_{rho }$$ and the generalized fractional maximal operators $$M_{rho }$$...     

Generalised weighted composition operators on Maeda–Ogasawara spaces

Every Archimedean Riesz space can be embedded as an order dense subspace of some $C^{\infty }\left(X\right)$, the Riesz space of all extended continuous functions on a Stonean space $X$, called its Maeda–Ogasawara space. Furthermore, it is a fact that every Riesz homomorphism between spaces of ordinary continuous functions on compact Hausdorff spaces is a weighted composition operator. We prove that a generalised statement holds for Maeda–Ogasawara spaces and refine these results in case the homomorphism preserves order limits.     

Nonclassical weighted estimates of certain Calderónzygmund operators on the plane

Let F be a compact subset of ? let μ be a Borel measure on F, and let ρ(z) be the distance of z to F. Denote $$A_K (f)(z) = \int\limits_F {K(\varsigma ,z)f(\varsigma ) dm(\varsigma ), z \in \mathbb{C}\backslash F}$$ where K (ζ, z) is either (ζ-z)² or (|ζ-z|(ζ-z))^-1 and m is the Lebesgue measure. Let ψ be a monotone nondecreasing positive function on (0, ∞) and let Φ(z)=Ψ(ρ(z))ρ(z), z ε ?/F. Under some additional assumptions on μ, it is proved that A_K is bounded from L² (μ) to L² (Φm) if and only if $$\int\limits_0^{ + 1} {\tfrac{{\psi (t)}}{t} + \int\limits_1^\infty {\tfrac{{\psi (t)}}{{t^2 }}dt< \infty } }$$ Thus, no interference of values of K of various signs is observed in such a situation. Bibliography: 4 titles.     

Legendre–Fenchel transform of the spectral exponent of polynomials of weighted composition operators

For the spectral radius of weighted composition operators with positive weight e ^φ T _α , \({\varphi\in C(X)}\) , acting in the spaces L ^p (X, μ) the following variational principle holds

$\ln r(e^\varphi T_\alpha)=\max_{\nu\in M^1_\alpha} \left\{\int\limits_X\varphi d\nu-\frac{\tau_\alpha(\nu)}{p}\right\},$

where X is a Hausdorff compact space, \({\alpha:X\mapsto X}\) is a continuous mapping and τ _α some convex and lower semicontinuous functional defined on the set \({M^1_\alpha}\) of all Borel probability and α-invariant measures on X. In other words \({\frac{\tau_\alpha}{p}}\) is the Legendre– Fenchel conjugate of ln r(e ^φ T _α ). In this paper we consider the polynomials with positive coefficients of weighted composition operator of the form \({A_{\varphi, {\bf c}}= \sum_{k=0}^n e^{c_k} (e^{\varphi} T_{\alpha})^k}\) , \({{\bf c}=(c_k)\in {\Bbb R}^{n+1}}\) . We derive two formulas on the Legendre–Fenchel transform of the spectral exponent ln r(A _φ,c) considering it firstly depending on the function φ and the variable c and secondly depending only on the function φ, by fixing c.

     

A class of normal weighted composition operators on the Fock space of ℂ n

This paper characterizes a class of normal weighted composition operators and their spectrum on the Fock space of Cⁿ.     

Interpolation theorems on weighted Lorentz martingale spaces

In this paper several interpolation theorems on martingale Lorentz spaces are given.The proofs are based on the atomic decompositions of martingale Hardy spaces over weighted measure spaces.Applying the interpolation theorems,we obtain some inequalities on martingale transform operator.     

Volterra-type operators from weighted Bergman–Orlicz space to $$\beta $$-Zygmund–Orlicz and $$\gamma $$γ-Bloch–Orlicz spaces

We find the conditions to ensure the boundedness and compactness of the Volterra-type operators acting from weighted Bergman–Orlicz space to \(\beta \)-Zygmund–Orlicz and \(\gamma \)-Bloch–Orlicz spaces, respectively.     

Maximal distributional chaos of weighted shift operators on Köthe sequence spaces

During the last ten some years, many research works were devoted to the chaotic behavior of the weighted shift operator on the Köthe sequence space. In this note, a sufficient condition ensuring that the weighted shift operator B _w ⁿ : λ _p (A) → λ _p (A) defined on the Köthe sequence space λ _p (A) exhibits distributional ?-chaos for any 0 < ? < diamλ _p (A) and any n ∈ ? is obtained. Under this assumption, the principal measure of B _w ⁿ is equal to 1. In particular, every Devaney chaotic shift operator exhibits distributional ?-chaos for any 0 < ? < diam λ _p (A).     

Estimates for the Upper Bounds of the First Eigenvalue on Submanifolds

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On Disjointly Homogeneous Orlicz–Lorentz Spaces

Mathematical Notes - A characterization of disjointly homogeneous Orlicz–Lorentz function spaces $$\Lambda_{\varphi,w}$$ is obtained. It is used to find necessary and sufficient conditions...     

Imbeddings between Weighted Orlicz–Lorentz Spaces

We establish necessary and sufficient conditions for imbeddings of weighted Orlicz–Lorentz spaces.     

Asymmetric forward–backward–adjoint splitting for solving monotone inclusions involving three operators

In this work we propose a new splitting technique, namely Asymmetric Forward–Backward–Adjoint splitting, for solving monotone inclusions involving three terms, a maximally monotone, a cocoercive and a bounded linear operator. Our scheme can not be recovered from existing operator splitting methods, while classical methods like Douglas–Rachford and Forward–Backward splitting are special cases of the new algorithm. Asymmetric preconditioning is the main feature of Asymmetric Forward–Backward–Adjoint splitting, that allows us to unify, extend and shed light on the connections between many seemingly unrelated primal-dual algorithms for solving structured convex optimization problems proposed in recent years. One important special case leads to a Douglas–Rachford type scheme that includes a third cocoercive operator.     

Note on conjectures of Beilinson–Bloch–Kato¶for cycle classes

Conjecures of Beilinson and Bloch–Kato describe the order of zeros of L-functions of motives in terms of motivic cohomology groups and Selmer groups. We restrict our attention to the parts generated by the cycle classes, and give modest evidence for the conjectures. Received: 7 June 1999 / Revised version: 7 September 1999